The term RPC typically refers to the Rational Polynomial Coefficient, or Rational Polynomial Camera coefficient [Chen et al., 2006]. It sometimes has been more generically defined as Rapid Positioning Capability [Dowman and Tao, 2002]. RPCs are sometimes also referred to as Rational Function Coefficients (RFCs), or Rational Functional Models (RFM) [Tao and flu, 2001]. RPCs are recommended by the OGC (Open GIS Consortium) and are widely used in the processing of high-resolution satellite images. An RPC model is a mathematical function that relates object space coordinates (latitude, longitude, and height) to image space coordinates (line and sample). It is expressed in the form of a ratio of two cubic functions of object space coordinates. Separate rational functions are used to express the object space to line, and the object space to sample, coordinate relationships [Dial and Grodecki, 2002a].
Because of ephemeris and attitude error, all satellite geometric sensor models, including physical and RPC models, have a definite value of absolute positioning error. For example, the ephemeris and attitude accuracy for IKONOS is about one meter for ephemeris and about one or two arc-seconds for attitude [Grodecki and Dial, 2003]. The accuracy for a single stereo pair of IKONOS images, without ground control, is 25.0 m (CE90), and 22.0 m (LE90) [Grodecki, 2001]. If the satellite positioning accuracy does not meet the needs of users, the sensor model should be refined by using Ground Control Points (GCPs) or other assistant data. Before the advent of IKONOS, users of satellite imagery typically made use of physical sensor models. Nowadays, instead of physical parameters, sometimes only a rational polynomial function which consists of 80 coefficients is available. This represents a completely new challenge, because the RPC has a high number of coefficients and there is no physical interpretation for the order and terms of these coefficients. Many researchers have attempted to address this challenge. Directly calculating a new RPC based on a large number of GCPs [Di et al., 2003] has been proven unfeasible [Grodecki and Dial., 2003; Hu et al., 2004]. The Batch Iterative Least-Squares (BILS) method and the Incremental Discrete Kalman Filtering (IDKF) method each requires a significant number of GCPs and also the covariance matrices of the RFCs, which are not available to most users [Hu and Tao, 2002]. The Pseudo GCP (PG) method, the Using Parameters Observation Equation (UPOE) method, and the Sequential Least Square Solution (SLSS) method [Bang et al., 2003] all face the problem of how to define the weightings of the coefficients for different observation equations.
In terms of accuracy and computational stability, the Bias Compensation method [Fraser and Hanley, 2003] so far appears to be the best method and has been widely used [Fraser and Hanley, 2003, 2005; Hu et al., 2004], but this method is effective only when the camera Field Of View (FOV) is narrow and the position and attitude errors are small [Grodecki and Dial, 2003]. Some satellites do meet these rigid conditions. For example as noted above, IKONOS imagery has an accuracy of about one meter for ephemeris and about one or two arc-seconds for attitude, and its FOV is less than one degree [Grodecki and Dial, 2003]. But many other satellites, including some of those launched from China and India, probably do not satisfy this condition. As a Generic Sensor Model (GSM), an RPC can accommodate an extremely wide range of images without a need for the satellite ephemeris [Samadzadegan et al., 2005]. Therefore, an RPC can be used in a number of different sensors, such as linear push-broom scanners, RADAR, airborne and space borne sensors. In these cases, the issue becomes one of how to effectively refine RPC using as few GCPs as possible.
On Sep. 24, 1999, IKONOS was launched. Since then, the mapping community has begun to recognize the importance of RPC; a mathematical function which relates the object space and image space (Equations 1 to 2).
                    x        =                                            P              1                        ⁡                          (                              X                ,                Y                ,                Z                            )                                                          P              2                        ⁡                          (                              X                ,                Y                ,                Z                            )                                                          (                  Eq          .                                          ⁢          1                )                                y        =                                            P              3                        ⁡                          (                              X                ,                Y                ,                Z                            )                                                          P              4                        ⁡                          (                              X                ,                Y                ,                Z                            )                                                          (                  Eq          .                                          ⁢          2                )                                          P          ⁡                      (                          X              ,              Y              ,              Z                        )                          =                              ∑                          i              =              0                                      m              ⁢                                                          ⁢              1                                ⁢                                          ⁢                                    ∑                              j                =                0                                            m                ⁢                                                                  ⁢                2                                      ⁢                                                  ⁢                                          ∑                                  k                  =                  0                                                  m                  ⁢                                                                          ⁢                  3                                            ⁢                                                          ⁢                                                a                  ijk                                ⁢                                  X                  i                                ⁢                                  Y                  j                                ⁢                                  Z                  k                                                                                        (                  Eq          .                                          ⁢          3                )                                                      0            ≤                          m              1                        ≤            3                    ;                ⁢                                  ⁢                              0            ≤                          m              2                        ≤            3                    ;                ⁢                                  ⁢                              0            ≤                          m              3                        ≤            3                    ;                ⁢                                  ⁢                                            m              1                        +                          m              2                        +                          m              3                                ≤          3                                    (                  Eq          .                                          ⁢          4                )            
Here (x, y) are the image coordinates, (X, Y, Z) are the ground coordinates, and aijk is the polynomial coefficient. One of the coefficients in the denominator is a constant 1. In some cases (e.g., IKONOS), the two denominators P2 and P4 have the same coefficients.
The RPC may be refined directly or indirectly. Direct refining methods modify the original RPCs themselves, while indirect refining methods introduce complementary or concatenated transformations in image or object space, and do not change the original RPCs directly [Hu et al., 2004].